Geometry & Topology

A very special EPW sextic and two IHS fourfolds

Maria Donten-Bury, Bert van Geemen, Grzegorz Kapustka, Michał Kapustka, and Jarosław Wiśniewski

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We show that the Hilbert scheme of two points on the Vinberg K3 surface has a two-to-one map onto a very symmetric EPW sextic Y in 5. The fourfold Y is singular along 60 planes, 20 of which form a complete family of incident planes. This solves a problem of Morin and O’Grady and establishes that 20 is the maximal cardinality of such a family of planes. Next, we show that this Hilbert scheme is birationally isomorphic to the Kummer-type IHS fourfold X0 constructed by Donten-Bury and Wiśniewski [On 81 symplectic resolutions of a 4–dimensional quotient by a group of order 32, preprint (2014)]. We find that X0 is also related to the Debarre–Varley abelian fourfold.

Article information

Geom. Topol., Volume 21, Number 2 (2017), 1179-1230.

Received: 28 September 2015
Revised: 28 January 2016
Accepted: 3 March 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14D06: Fibrations, degenerations 14J35: $4$-folds 14J70: Hypersurfaces 14K12: Subvarieties 14M07: Low codimension problems
Secondary: 14J50: Automorphisms of surfaces and higher-dimensional varieties 14J28: $K3$ surfaces and Enriques surfaces

EPW sextics IHS fourfolds abelian varieties


Donten-Bury, Maria; van Geemen, Bert; Kapustka, Grzegorz; Kapustka, Michał; Wiśniewski, Jarosław. A very special EPW sextic and two IHS fourfolds. Geom. Topol. 21 (2017), no. 2, 1179--1230. doi:10.2140/gt.2017.21.1179.

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